📂Complex Anaylsis

The Relationship between Trigonometric Functions and Exponential Functions in Complex Analysis

Theorem ¹

The sine, cosine functions as complex functions $\sin , \cos : \mathbb{C} \to \mathbb{C}$ are as follows. $$\sin z = { {e^{iz} - e^{-iz}} \over 2 i } \\ \cos z = { {e^{iz} + e^{-iz}} \over 2 }$$

Description

It’s actually okay to think of this more as a definition than a theorem. The purpose is to demonstrate that defining it this way does not conflict with theorems that have already been established. The proof is merely a reorganization of what we already knew from Euler’s formula, tailored to trigonometric functions.

Proof

By Euler’s formula $\displaystyle { e }^{ ix }= \cos x + i \sin x$, $$\begin{cases} { e }^{ iz }= \cos z + i \sin z \\ { e }^{ -iz }= \cos z - i \sin z \end{cases}$$ rearranging these for trigonometric functions gives us $$\sin z = { {e^{iz} - e^{-iz}} \over 2 i } \\ \cos z = { {e^{iz} + e^{-iz}} \over 2 }$$

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